Bravais lattice table

When these four types of lattice are combined with the 7 possible unit cell shapes, 14 permissible Bravais lattices (Table 1.3) are produced. (It is not possible to combine some of the shapes and lattice types and retain the symmetry requirements listed in Table 1.2. For instance, it is not possible to have an A-centred, cubic, unit cell if only two of the six faces are centred, the unit cell necessarily loses its cubic symmetry.)... 

Table 5.2 contains the results of measurements made on the powder ditfracto-grams of four cubic substances obtained with CuKa radiation of wavelength 1= 1.5418 A, The Bragg angles 6 were measured with a precision of 0.0 T the intensities I are normalized such that I = 100 for the strongest line. Find the indices hU of the reflectiom and thfiiattice constants (Section 3.5.3). For each compound, identify the systematic absences and deduce the Bravais lattice (Table 3.2). 

If atoms, molecules, or ions of a unit cell are treated as points, the lattice stmcture of the entire crystal can be shown to be a multiplication ia three dimensions of the unit cell. Only 14 possible lattices (called Bravais lattices) can be drawn in three dimensions. These can be classified into seven groups based on their elements of symmetry. Moreover, examination of the elements of symmetry (about a point, a line, or a plane) for a crystal shows that there are 32 different combinations (classes) that can be grouped into seven systems. The correspondence of these seven systems to the seven lattice groups is shown in Table 1. 

These 14 Bravais Lattices are unique in themselves. If we arrange the crystal systems in terms of symmetry, the cube has the highest symmetry and the triclinic lattice, the lowest symmetry, as we showed above. The same hierarchy is maintained in 2.2.4. as in Table 2-1. The symbols used by convention in 2.2.4. to denote the type of lattice present are... 

Compared with the problem in three-dimensional space, which has 14 Bravais lattices and 230 space groups, the problem of surface symmetry is tmly a dwarf It only has 5 Bravais lattices and 17 different groups. The five Bravais lattices are listed in Table E.l. 

It is possible to characterize the type of Bravais lattice present by the pattern of systematic absences. Although our discussion has centred on cubic crystals, these absences apply to all crystal systems, not just to cubic, and are summarized in Table 2.3 at the end of the next section. The allowed values of are listed in Table 2.2 for... 

X-ray powder data for NaCl is listed in Table 2.5. Determine the Bravais lattice, assuming that it is cubic. 

A.2 The fourteen Bravais lattices and seven crystal systems Refer to Figs. A.2.1 and Table A.2.1. 

Table 1.1 gives the structures of the elements at zero temperature and pressure. Each structure type is characterized by its common name (when assigned), its Pearson symbol (relating to the Bravais lattice and number of atoms in the cell), and its Jensen symbol (specifying the local coordination polyhedron about each non-equiyalent site). We will discuss the Pearson and Jensen symbols later in the following two sections. We should note,... 

Local coordination polyhedra the Jensen notation 7 Table 1.2 The fourteen Bravais lattices in three dimensions... 

Table 17.1. Irreducible representations and basis functions for the symmetry point X in the BZ of the sc Bravais lattice.

Table 17.13. Free-electron band energies em for InSb space group 216) at T, A, and X, the BZ of the fee Bravais lattice (Figure 16.12(b)).

Chapter 2 Crystals, Point Groups, and Space Groups Table 2.1. Crystal systems and the 14 Bravais Lattices. 

The fourteen Bravais lattices are divided into seven crystal systems. The term system indicates reference to a suitable set of axes that bear specific relationships, as illustrated in Table 9.2.1. For example, if the axial lengths take arbitrary values and the interaxial angles are all right angles, the crystal system... 

Bravais then showed that in three dimensions, there are only 14 different lattice types, currently named the Bravais lattices, which are grouped in seven crystal systems [1-3] (see Table 1.1). 

In Table 1.2, the subtypes corresponding to each crystal system are listed and in Figure 1.4, the 14 Bravais lattices in three dimensions are illustrated. 

Among the 14 cells that generate the Bravais lattices (see Figure 1.4), only the P-type cells are considered primitive unit cells. It is possible to generate the other Bravais lattices with primitive unit cells. However, in practice, only unit cells that possess the maximum symmetry are chosen (see Figure 1.4 and Table 1.2) [1-6]. 

The 32 crystallographic point groups, first mentioned in Table 7.1, are now described in Table 7.8 (ordered by principal symmetry axes and also by the crystal system to which they belong). The 230 space groups of Schonflies and Fedorov were generated systematically by combining the 14 Bravais lattices with the intra-unit cell symmetry operations for the 32 crystallographic point... 

The 14 Bravais lattices are enumerated in Table 9-4 as the following types primitive (P, R), side-centered (C), face-centered (F), and body-centered (7). The numbering of the Bravais lattices in Table 9-4 corresponds to that in Figure 9-20. The lattice parameters are also enumerated in the table. In addition, the distribution of lattice types among the crystal systems is shown. 

TABLE 4.3. The 14 Bravais Lattices, 32 Crystallographic Point Groups (Crystal Classes) and Some Space Groups. 

The introduction of lattice centering makes the treatment of crystallographic symmetry much more elegant when compared to that where only primitive lattices are allowed. Considering six crystal families Table 1.12) and five types of lattices Table 1.13), where three base-centered lattices, which are different only by the orientation of the centered faces with respect to a fixed set of basis vectors are taken as one, it is possible to show that only 14 different types of unit cells are required to describe all lattices using conventional crystallographic symmetry. These are listed in Table 1.14, and they are known as Bravais lattices. ... 

The international crystallographic space group symbols begin with a capital letter designating Bravais lattice, i.e. P, A, B, C, I, F or R (see Table 1.13 and Table 1.14). 

First, consider the Bravais lattices (see Table 1.14) and point groups (see Table 1.8) which are allowed in a given crystal system. 

For example, think about the monoclinic point group m in the standard setting, where m is perpendicular to b (Table 1.8). According to Table 1.14, the following Bravais lattices are allowed in the monoclinic crystal system P and C. There is only one finite symmetry element (mirror plane m) to be considered for replacement with glide planes (a, b, c, n and d) ... 

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